ieee transactions on wireless communications, vol. x,...

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. Y, SEPTEMBER 2018 1

Sparse Vector Coding for Ultra Reliable and LowLatency Communications

Hyoungju Ji, Member, IEEE, Sunho Park, Member, IEEE, and Byonghyo Shim, Senior Member, IEEE

Abstract—Ultra reliable and low latency communication(URLLC) is a newly introduced service category in 5G to supportdelay-sensitive applications. In order to support this new servicecategory, 3rd Generation Partnership Project (3GPP) sets anaggressive requirement that a packet should be delivered with10−5 packet error rate within 1 ms transmission period. Since thecurrent wireless transmission scheme designed to maximize thecoding gain by transmitting capacity achieving long codeblockis not relevant for this purpose, a new transmission schemeto support URLLC is required. In this paper, we propose anew approach to support the short packet transmission, calledsparse vector coding (SVC). Key idea behind the proposedSVC technique is to transmit the information after the sparsevector transformation. By mapping the information into theposition of nonzero elements and then transmitting it after therandom spreading, we obtain an underdetermined sparse systemfor which the principle of compressed sensing can be applied.From the numerical evaluations and performance analysis, wedemonstrate that the proposed SVC technique is very effectivein URLLC transmission and outperforms the 4G LTE and LTE-Advanced scheme.

Index Terms—Short packet transmission, support identifica-tion, 5G, ultra reliable and low latency communications.

I. INTRODUCTION

Ultra reliable and low latency communication (URLLC) isa newly introduced service category in 5G to support delay-sensitive applications such as the tactile internet, autonomousdriving, factory automation, cyber-physical system, and remoterobot surgery [2]. In order to support this new service category,3rd Generation Partnership Project (3GPP) sets an aggressiverequirement that a packet should be delivered with 10-5 blockerror rate (BLER) within 1 ms period [3]. One notable ob-servation in these applications is that the transmit informationis control (command) type information (e.g., move left/right,start/stop, rotate/shift, and speed up/down) or sensing infor-mation (e.g., temperature, moisture, pressure, and gas density)so that the amount of information to be delivered is tiny [4].Since the current wireless transmission strategy designed tomaximize the coding gain by transmitting capacity achievinglong codeblock is not relevant to these URLLC scenarios,

H. Ji, S. Park, and B. Shim are with the Institute of New Media andCommunications and the Department of Electrical and Computer Engineer-ing, Seoul National University, Korea (e-mail: [email protected],[email protected], [email protected]).

A part of this work has been presented in Information Theory andApplications (ITA) Workshop, San Diego, USA, Feb. 11-16, 2018 [1].

This work was sponsored by the National Research Foundation of Korea(NRF) grant funded by the Korean government(MSIP) (2016R1A2B3015576)and by ’The Cross-Ministry Giga KOREA Project’ grant funded by the Koreagovernment(MSIT) (No.GK17P0500, Development of Ultra Low-LatencyRadio Access Technologies for 5G URLLC Service).

entirely new transmission strategy to support the short packettransmission is required. While there have been some effortsto improve the connection density, the medium access latency,and the reliability of the re-transmission scheme for URLLC[5–9], not much work has been done for the short-sized packettransmission except for the consideration of advanced channelcoding schemes (e.g., polar code) [10].

In the current 4G systems, reliability of the data trans-mission is mainly achieved by the channel coding scheme[11]. Encoding at the basestation is done by the convolutioncoding and the decoding at the mobile terminal is done bythe maximum likelihood decoding (MLD) or Turbo decoding.While this approach has shown to be effective in 4G systems,use of this scheme in URLLC scenario would be problematicsince there is a stringent limitation on the packet length (andthus the parity size) yet the required reliability (target BLER= 10-5) is much higher than the current LTE-Advanced andLTE-Advanced Pro systems (target BLER = 10-2∼10-3) [10].

The purpose of this paper is to propose a new type of shortpacket transmission for URLLC that does not rely on theconventional channel coding principle. Key idea behind theproposed technique, henceforth referred to as sparse vectorcoding (SVC), is to transmit the short-sized information afterthe sparse vector transformation. To be specific, by mappingthe information into the sparse vector and then transmittingit after the random spreading, we obtain an underdeterminedsparse system for which the principle of compressed sensingcan be applied [12]. It is now well-known from the theoryof compressed sensing that an accurate recovery of a sparsevector is guaranteed with a relatively small number of mea-surements when the system matrix (a.k.a. sensing matrix)is generated at random [13], which is achieved in our casevia the random spreading. In fact, since the sparsity of theinput vector is guaranteed by the sparse vector transformation,SVC decoding is achieved by the sparse signal recovery(more accurately, identification of nonzero positions in thetransmit sparse vector). Therefore, the proposed scheme is verysimple to implement and can be applied to wide variety offuture wireless applications in which the amount of transmitinformation is sufficiently small.

We note that there have been various efforts to use thesupport locations in the information encoding process [14–18]. For example, sparse mapping is conceptually similarto the position modulation (PM) and the index modulation(IM) techniques [14–16] in which the indices of the buildingblock of the communication systems, such as pulses in opticalsystems, transmit antennas at the basestation or subcarriergroups in OFDM systems, are used to convey additional infor-


mation bits. Also, in the single and multiple PM techniques,information is transmitted via the time sparsity by using thecombinations of the positions of optical pulses. In the spatialmodulation-based IM technique, for example, additional infor-mation can be delivered by selectively using part of transmitantennas in the information transmission. Similar approach canalso be found in Boolean multiple access channel [17-18]. Ourwork is distinct from these studies in that we fully utilize thephysical resources in the data transmission so that the loss, ifany, caused by the underutilization of physical resources canbe prevented. Also, in contrast to the IM technique where thereceiver processing consists of two steps (the index recoveryand symbol detection), decoding of the proposed SVC schemeis achieved by the identification of nonzero position calledsupport identification [12]. The distinction of SVC over theconventional techniques is further strengthened by the fact thatthere is no random sensing mechanism (e.g., random spreadingin SVC) in the conventional schemes so that the compressionof the transmit vector is not possible.

From the performance analysis in terms of the decod-ing success probability and also numerical evaluations onthe realistic setting, we demonstrate that the proposed SVCtechnique is very effective in short-size packet transmissionand outperforms the 4G LTE and LTE-Advanced physicaldownlink control channel (PDCCH) scheme by a large marginin terms of reliability and transmission latency.

The rest of this paper is organized as follows. In Section II,we briefly explain the short-sized packet transmission in 4GLTE and LTE-Advanced systems. In Section III, we present theproposed SVC scheme and explain the encoding and decodingoperations. In Section IV, we analyze the success probabilityof SVC-encoded data transmission. Various implementationissues are discussed in Section V. In section VI, we presentsimulation results to verify the performance of the proposedscheme. We conclude the paper in Section VII.

II. SHORT-SIZED PACKET IN LTE-ADVANCED DOWNLINK

In this section, we briefly review the control-type datatransmission (PDCCH of 4G LTE systems) to illustrate theshort-sized packet transmission in the conventional systems.PDCCH carries essential information for the mobile terminalwhen it tries to transmit or receive the data. To be specific,PDCCH carries small-sized information needed to decode thedata channel (e.g., resource assignment, modulation order,code rate). On top of these, cyclic redundancy check (CRC)is added to test the decoding error [19]. Since the CRC bitstream is scrambled with a user index (called radio networktemporary identifier), only the scheduled user can pass theCRC test.

After the channel coding and symbol mapping,1 the modu-lated symbol vector s ∈ CN×1 is transmitted. The correspond-ing received vector y ∈ Cm×1 is given by

y = HRs + v, (1)

1e.g., convolution coding with rate 13 and quadrature phase shift keying

(QPSK) modulation are employed.

1 0 0 0 1 0 0 0 0

Fig. 1: Metaphoric illustration of SVC encoding. Informationis mapped into the position of a sparse vector.

where H ∈ Cm×m is the diagonal matrix whose diagonalentry hii is the channel component for each resource, v ∼CN(0, σ2

v I) is the additive Gaussian noise, and R ∈ Cm×N isthe matrix describing the mapping between the symbol andresource element. For example, when one symbol is mappedto a single resource, R would be the identity matrix (R = I).Whereas, if two resources are assigned to one symbol for thetransmit diversity, then R would be 2N × N matrix (e.g., ifN = 2, then R =

[1 0 1 00 1 0 1

]T ).When one tries to improve the reliability with a small mod-

ification of current PDCCH, one can think of three options.The first option is to achieve the better coding gain by usinglower code rate (i.e., r = b

2N < rpdcch = 13 ). This option is

easy and straightforward but when the coded symbol lengthN increases, transmission and processing latency will alsoincrease, resulting in the violation of the URLLC requirement.The second option is to use the multiple resources to achievethe diversity gain (m > N). By combining multiple versionsof the same symbol at the receiver, reliability of the symbolcan be improved. The problem of this approach is that a largeportion of wireless resources are consumed in achieving thediversity gain so that there would be a severe degradation ofthe resource utilization efficiency. The third option is to reducethe size of control information b. By removing some of thescheduling parameters, resources used for the control channelcan be saved. Even in this case, it is not possible to removeessential information (e.g., CRC and user index) so that onecannot expect a dramatic reduction of control information.

III. SPARSE VECTOR CODING

A. SVC Encoding and Transmission

The key idea of the proposed SVC technique is to map theinformation into the positions of a sparse vector s. Figurativelyspeaking, SVC encoding can be thought as marking a fewdots to the empty table. As illustrated in Fig. 1, if we tryto mark dots to two cells out of 9, then there would be(92)= 36 choices in total. In general, when we choose K

out of N symbol positions, we can encode blog2(NK

)c bits

of information. Example of one-to-one mapping between theinformation bit stream w and transmit sparse vector s is (seeexample in Table I)

0 0 0 0 00 0 0 0 10 0 0 1 1

...1 1 1 1 1︸︷︷︸

b-bit information w(b=5)

←→

←→

←→...←→

0 0 0 0 0 0 0 1 10 0 0 0 0 0 1 0 10 0 0 0 0 1 0 0 1

...1 0 0 0 0 0 0 0 1︸︷︷︸K−sparse vector s (K=2)

.


TABLE I: Example of mapping between the information wand the sparse vector s

Input: (w)(10) sSize of sparse vector N , 0 000011information vector w 1 000101

Output: 2 000110Sparse vector s 3 001001

a := 0 4 001010for i = 2 to N do 5 001100for j = 1 to i − 1 do 6 010001if a = (w)(10) 7 010010s :=

(2i + 2 j

)(2) 8 010100

end if 9 011000a := a + 1 10 100001

end for :end for :

Note: (w)(10) is decimal expression of binary vector w and (w)(2)is binary expression of integer w.

After the sparse mapping, each nonzero element in s isspread into m resources using the codeword (spreading se-quence) in the spreading codebook C. While it is possibleto allocate resources either in time, frequency axis or hybridof these, in this work, we assume that they are allocated inthe frequency axis (see Fig. 2(a)). This choice will not affectthe system model but minimizes the transmission latency.As a result of this spreading process called the multi-codespreading, the resource mapping matrix R in (1) is replacedwith the codebook matrix C = [c1 c2 · · · cN ] where ci =[c1i c2i · · · cmi]

T is the spreading sequence. For example,if the first and the third element of s are nonzero, then thetransmit vector after spreading is

x = Cs= s1c1 + s3c3. (2)

Since the positions of nonzero elements are chosen at random,the codebook matrix C should be designed such that thetransmit vector x contains enough information to recover thesparse vector s irrespective of the selection of the nonzeropositions. It has been shown that if entries of the codebookmatrix C are generated at random, e.g., sampled from Gaussianor Bernoulli distribution, then an accurate recovery of thesparse vector is possible as long as m = O (K log N) [13].Example of C for m = 5 and N = 10, when elements of ciare chosen from the Bernoulli distribution, is given by

C =1α

1 1 1 1 −1 1 −1 1 −1 −11 −1 1 −1 1 −1 1 −1 −1 11 1 −1 −1 1 1 −1 −1 1 11 −1 −1 1 1 −1 1 1 −1 −1−1 1 1 1 −1 −1 −1 −1 1 1

,(3)

where α is the normalization factor depending on the modu-lated symbols (see Section V.B). The corresponding receivedsignal y is

y = Hx + v

=[Hc1 Hc3

] [s1s3

]+ v. (4)

time

frequency

time

frequency

PDCCH region

Pilot symbol

Pilot region

Control region

Data region

Control transmissionData transmission

URLLC transmission

m

(a)

Datainformation

Sparsemapping

b bits Multi-codespreading

Channel

Supportdetection

Sparsedemapping

N symbols

s

Decodedinformation

x = Cs

ys

Basestation encoding

Decoding at mobile station

(b)

Fig. 2: SVC-based packet transmission: (a) packet structureof 4G (left) and the URLLC packet (right) and (b) the blockdiagram for the proposed SVC technique.

In general, the received vector y is given by

y = HCs + v

=

h11

. . .

hmm

c1 . . . cN

s1...

sN

+v1...vm

. (5)

It is worth mentioning that an accurate recovery of the sparsevector s is unnecessary in SVC since the decoding of theinformation vector is achieved by the identification of nonzeropositions, not the actual values of this vector. The fact thatthe decoding is done by the support2 identification greatlysimplifies the decoding process and also reduces the chance ofdecoding failure. The overall structure of the proposed SVCis depicted in Fig. 2(b).

The benefits of SVC can be summarized as follows; First,the transmission power of the data channel is concentratedon the nonzero elements of an information vector. Thus, whencompared to the conventional system in which the transmissionpower is uniformly distributed across all symbols, effectivetransmit power per symbol is higher. Second, the SVC decod-ing process achieved by the sparse recovery algorithm lendsitself to the test of decoding success/failure so that the CRCoperation is unnecessary. This directly implies that the coderate of SVC can be made smaller than the rate of PDCCH.Specifically, when the number of resources used for the data

2Support is the set of nonzero elements. For example, if s = [0 0 1 0 0 1],the Ωs = 3, 6.


TABLE II: PDCCH versus SVC technique

PDCCH SVC technique

Coding(encod-ing/decoding)

Convolution code ( 13

rate) / Viterbi decodingSparse encoding / CS re-covery algorithm

Transmission Time/frequencymapping

Spreading in frequencydirection

User identifica-tion

CRC scrambled withuser index

User codebook C

Resource over-head (L repeti-tions, QPSK)

L 3b2 Lm where m is the size

of spreading length

channel is m and the QPSK modulation is used, the coderate of SVC is rsvc =

bi

2m (bi is the number of information

bits) and the code rate of PDCCH is rpdcch =(bi+bc )

2m

(= 1

3

).

If the number of CRC bits is bc = βbi (β > 0), thenm = 3

2 (bi + bc) = 32 (bi + βbi). Thus, the code rate of SVC

can be expressed in term of β as

rsvc =bi2m=

13(1 + β)

<13= rpdcch . (6)

Third, when m is sufficiently large, the basestation can easilyassign the distinct codebook C for each user. This is becausecodebook matrices can be made near orthogonal by usinga properly designed codebook generation mechanism.3 Forexample, when m = 42 and the codebook is generated by theBernoulli distribution, then there are 242 different spreadingsequences ci . Thus, if N = 96, then the basestation cansupport maximally 235(≈ 242

96 ) devices. Last but not leastimportant benefit of SVC is that the implementation cost issmall and the processing latency is low. Encoding is donevia a simple injective mapping and spreading, which can beeasily realized by the look-up table and addition/subtractionoperations and the decoding is performed by the supportdetection and demapping. In particular, since the sparsity Kis small and also known to the receiver, one can decode theSVC packet using a simple sparse recovery algorithm suchas orthogonal matching pursuit (OMP) [12].4 Comparisons ofPDCCH and SVC are summarized in Table II.

B. SVC Decoding

1) Support Identification: As mentioned, the SVC decodingis done by the identification of the support and any sparserecovery algorithm can be employed for this purpose. In thiswork, we employ the greedy sparse recovery algorithm in thedecoding of the SVC-encoded packet. After pre-multiplyingthe diagonal matrix constructed by the complex exponentiale j]h , the modified received vector can be expressed as

y = diag [exp( j]h11) . . . exp( j]hmm)] y= diag

[h]

Cs + v,

3The correlation between two distinct columns of random matrix decreasesexponentially as the dimension of a column increases (see, e.g., [20, Theorem1]).

4Most of CS algorithm finds out the solution without the prior knowledgeof the sparsity K . However, when K is known in advance, one can recover thesparse vector more accurately by using the sparsity-aware recovery technique[29].

= HCs + v, (7)

where ]h is the angle of h, h = [h11e j]h11 . . . hmme j]hmm ],H = diag

[h], and v = [v1, . . . , vm] is the modified noise

vector where vi = vie j]hii . Since s has K nonzero elements,the modified received vector y = HCs+ v can be expressed asa linear combination of K columns of Φ = HC perturbedby the noise. In view of this, the main task of the SVCdecoding is to identify the columns in Φ participating inthe modified received vector. In each iteration, greedy sparserecovery algorithm identifies one column of Φ at a time usinga greedy strategy [21]. Specifically, a column of Φ that ismaximally correlated with the (modified) observation rj−1 ischosen. That is, an index of the nonzero column of Φ chosenas j-th iteration is5

ωj = arg maxl|<φl, rj−1>|2, (8)

where rj−1 = y −ΦΩ

j−1s

sj−1 is the modified observation called

the residual and sj−1 = Φ†Ω

j−1s

y is the estimate of s at ( j −1)-thiteration.6

A better way to improve the decoding performance is touse the maximum likelihood (ML) detection. Recalling thatthe sparsity K is known to both transmitter and receiver, theML detection problem for the system model in (8) is

s∗ = arg max‖s‖0=K

Pr (y|s, H,C), (9)

where ‖s‖0 is the `0-norm of s counting the number of nonzeroelements in s. Since our goal is to find out the support of s,we alternatively have

Ω∗s = arg max

|Ωs |=KPr (y|Ωs, H,C), (10)

where |Ωs | is the cardinality of the set Ωs.To find out the ML solution, we need to enumerate all

possible combinations of candidate supports with cardinalityK . Unfortunately, this exhaustive search would not be feasiblefor most practical scenarios. In this work, we instead use themultipath match pursuit (MMP) algorithm [22], a recently pro-posed near-ML sparse recovery algorithm, as a baseline for theSVC decoding. In a nutshell, MMP performs an efficient treesearch to find out the near-ML solution to the original sparsevector. Unlike the single-path search algorithm, MMP selectsmultiple promising indices in each iteration. Specifically, eachcandidate chosen in an iteration brings forth multiple new childcandidates. After finishing K iterations, candidate s∗ having thesmallest cost function among all candidates is chosen as thefinal output (i.e., s∗ = arg min

sJ(s) where J(s) = ‖y −ΦΩs s‖2).

Due to the fact that many candidates are redundant and hencecounted only once, an actual number of candidates examinedin MMP are quite moderate [22].

One clear advantage of MMP, in the perspective of SVCdecoding, is that it deteriorates the quality of incorrect candi-date yet does not impose any estimation error to the correctone. This is because the quality of incorrect candidates gets

5If Ω = 1, 3, then ΦΩ = [φ1 φ3].6Φ† = (ΦT Φ)−1ΦT is the pseudo-inverse of Φ.


TABLE III: The proposed MMP-based SVC decoding algo-rithm

Input:Measurement y, sensing matrix Φ = HC, sparsity K ,number of expansion L, max number of search candidate lmax ,stop threshold ε , detection threshold ε

Output:Support set Ω

Initialization:l := 0 (candidate order), ρ := ∞ (minimum magnitude of residual)

While: l < lmax and ε < ρ dol := l + 1r0 := y[p1, ... , pK ] := compute_pk (l, L) (compute layer order)for k = 1 to K do (investigate l-th candidate)ω :=compute_ω(k, L) (choose L best indices)Ωkl

:= Ωk−1l∪ ωpk

(construct a path in k-th layer)rk := y −Φ

Ωkl

sk (update residual)

Ωk := Ωkl

(update support set)end forif ‖rK ‖22 < ρ then (update the smallest residual)ρ := ‖rK ‖22

if‖rK ‖22‖y‖22

> 1− ε then (false-alarm identification)

Ω∗ := 0end ifΩ∗ := ΩK

end ifend whilereturn Ω∗function compute_pk (l, L)t := l − 1for k = 1 to K dopk := mod (t, L) + 1t := floor(t/L)

end for return [p1, ... , pK ]end functionfunction compute_ω(k, L)if k = odd thenreturn arg max

|π |=L‖(<〈

φT

‖φ‖2rk−1 〉)π ‖22

elsereturn arg max

|π |=L‖(=〈

φT

‖φ‖2rk−1 〉)π ‖22

end ifend function

worse due to the error propagation while no such behavioroccurs to the correct one. In particular, since nonzero valuesof an original sparse vector s are known to the receiver,7

no estimation error will be introduced in the correct candi-date. We note that the computational complexity of the SVCdecoding is marginal since the computational complexity ofthe greedy sparse recovery algorithm is directly proportionalto the sparsity K .8 Accordingly, the processing latency ofSVC decoding can also be made sufficiently small. This is incontrast to the Viterbi or Turbo decoding algorithm in whichthe computational complexity is proportional to the length ofa codeblock [23].

7Since the goal of SVC decoding is to find out the nonzero positions ofa sparse vector, we can pre-define values of the nonzero elements in s (seeSection IV.A).

8In each iteration, greedy sparse recovery algorithm performs three opera-tions: support identification, nonzero element estimation, and residual update.Since the nonzero values are fixed and known in advance, estimation of thenonzero elements is unnecessary.

0 1 2 3 40

20

40

60

80

100

k

‖rk‖

2 2/‖y‖

2 2(%

)

y = v

y = HC′x + v (C′ , C)

y = HCx + v

Fig. 3: Snapshot of the ratio between residual magnitude ‖rk ‖22and ‖y‖22 as a function of the number of iterations in the OMPalgorithm. Signal-to-noise ratio (SNR) is set to 0 dB and thesparsity K is set to 4.

2) Identification of False Alarm: Overall, there are twokinds of false alarm events causing the decoding failure: 1)support detection when the basestation transmits informationto the different user and 2) support detection when thereis no transmission at the basestation. In order to preventthese events, we need to examine the residual magnitudein each iteration. Firstly, when a packet for the differentuser is received, the codebook between two distinct userswould be different from each other so that the magnitudeof the correlation µi j between two codewords, each beingchosen from two district codebooks would be small. In thiscase, clearly, one cannot expect a substantial reduction in theresidual magnitude. Secondly, when there is no transmission,the received vector will measure the noise only (i.e., y = v)and thus some column in Φ, say φl , will be added to theresidual in each iteration ri = ri−1 − φl sl (see Fig. 3). Basedon these observations, we declare the decoding failure whenthe residual magnitude is outside of the confidence intervalof the pure noise contribution. We will say more about theselection of confidence interval Section V.E.

The proposed MMP-based SVC decoding algorithm is sum-marized in Table III.

IV. SVC PERFORMANCE ANALYSIS

In this section, we analyze the decoding success probabilityof the SVC technique. As mentioned, decoding of the SVC-encoded packet is successful when all support elements arechosen by the sparse recovery algorithm so that we analyzethe probability that the support is identified accurately. In ouranalysis, we assume that the greedy sparse recovery algorithmis used in the decoding process and analyze the lower boundof the success probability. For analytic simplicity, we initiallyconsider K = 2 scenario and then extend to the general case.Without loss of generality, we assume that p and q-th elementsof s are nonzero (i.e., Ωs = p, q). Further, by setting the


information vector such that sp = 1 and sq = j, we can modelthe QPSK transmission (see Section V.B).

Following lemmas will be useful in our analysis.

Lemma 1. Consider the vector ai (i = 1, · · · , N) whoseelement is i.i.d. standard Gaussian. Then,

aTi a j

‖ai ‖2 is standard

Gaussian. That is,aTi a j

‖ai ‖2 ∼ N(0, 1).

Proof: See Appendix A.

Lemma 2. Consider the vector h = [h11 h22 · · · hmm]T where

hii = hiie j]hii . The probability density function (PDF) of the‖h‖22 is Chi-squared distribution with

f‖h‖22 (x) =xm−1 exp (−x)Γ(m)

, (11)

where Γ(m) = (m−1)! is the Gamma function and E[‖h‖22

]=

m.

Proof: From (7), ‖h‖22 can be expressed as ‖h‖22 = ‖h‖22 =∑m

i=1 |hii |2 =

∑mi=1(<(hii)

2+=(hii)2) where <(c) and =(c) arethe real and imaginary part of c, respectively. Since <(hii),=(hii) ∼ N(0,

σ2v

2 ), we can show after some manipulationsthat 2‖h‖22 follows Chi-squared distribution with 2m DoF [28].That is,

f2‖h‖22 (x) =xm−1 exp

(− x

2)

2mΓ(m). (12)

Since fZ (z) = 2 f2Z (2z), we have

f‖h‖22 (x) =xm−1 exp (−x)Γ(m)

. (13)

Let S j be the success probability that the support element ischosen in the j-th iteration. Since K = 2 and thus the requirednumber of iterations to decode the information vector is two,the probability that the SVC packet is successfully decodedcan be expressed as

Psucc = P(Ω∗s = Ωs)

= P(S1,S2

)= P

(S2 |S1

)P

(S1

). (14)

Our main result in this section is as follows.

Theorem 1. The probability that the SVC-encoded packet isdecoded successfully satisfies

Psucc ≥

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m)2N

,(15)

where m is the number of measurements (resources), N isthe size of sparse vectors, σ2

v is the noise variance, andµ∗ = max

i,j|µi j | is the maximum absolute value of correlation

between two distinct columns of Φ.

When m is sufficiently large, we approximately have

Psucc &

(1 −

(1 +(1 − µ∗)2

σ2v

)−m)2N

.(16)

Also, since the block error rate is BLERsvc = 1 − Psucc , theupper bound of BLER is

BLERsvc . 1 −(1 −

(1 +(1 − µ∗)2

σ2v

)−m)2N

.(17)

In Fig. 4, we plot the BLER performance of SVC as afunction of SNR. To judge the effectiveness of Theorem 1,we perform the empirical simulation for m = 42, N = 96.From the empirical evaluations, we obtain that µ∗ ≈ 0.7. Whenwe apply this value to the upper bound in (17), we couldobserve that the obtained bound is tight across the board. Tobetter understand the performance of SVC, we plot the BLERas a function of µ∗, N, and m in Fig. 4(b), 4(c), and 4(d).First, when the maximum correlation µ∗ decreases, we seethat the BLER gain increase sharply as shown in Fig. 4(b).For example, if µ∗ is reduced from 0.4 to 0.2, we can achieve1.5 dB gain at the target reliability point (BLER = 10-5).Next, we test the BLER performance for various sparse vectordimensions in Fig. 4(c). Although the BLER performancedegrades with N , we see that the degradation is fairly graceful.Whereas, as shown in Fig. 4(d), the BLER performance isquite sensitive to the number of measurements.

As a first step to prove Theorem 1, we analyze the successprobability P

(S1) for the first iteration.

Lemma 3. Consider the received signal y = γΦs + vwhere γ =

√SNRα , Φ = [φ1 φ2 · · · φN ], and φi =

[h11c1i h22c2i · · · hmmcmi]T . The probability that the support

element is chosen in the first iteration satisfies

P(S1) ≥

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m)N−1

.(18)

Proof: As shown in Table III, N decision statisticsφT

l

‖φl ‖2rk−1 (l = 1, · · · , N) are computed in each iteration. For

analytic simplicity, we take the real part of the decision statisticin the first iteration and the imaginary part in the seconditeration.9

In order to identify the support element in the first iteration,we should have

<〈 φp

‖φp ‖2, r0〉

≥ maxi

<〈 φi

‖φi ‖2, r0〉

and thusthe success probability for a given channel realization h is

P(S1 |h) = P(<〈 φp

‖φp ‖2, r0〉

≥ maxi

<〈 φi

‖φi ‖2, r0〉

)=

N∏i=1,i,p

P(<〈 φp

‖φp ‖2, r0〉

≥ <〈 φi

‖φi ‖2, r0〉

) ,(19)

where 〈a, b〉 is the inner product between two vector a and b.First, noting that sp = 1 and sq = j, we have

〈φp

‖φp ‖2, r0〉 = 〈

φp

‖φp ‖2,φpsp + φqsq + v〉

= ‖h‖2 + j‖h‖2µqp +φTl

‖φl ‖2v, (20)

9This choice is suboptimal but simplifies the analysis.


0 2 4 6 810−5

10−4

10−3

10−2

10−1

1

SNR (dB)

BL

ER

SimulationUpper bound in (17)

-4 -2 0 2 4 610−5

10−4

10−3

10−2

10−1

1

SNR (dB)

BL

ER

µ∗ = 0.0µ∗ = 0.2µ∗ = 0.4µ∗ = 0.6µ∗ = 0.8

(a) the exact BLER and bound (m = 42, N = 96) (b) BLER with µ∗ (m = 42, N = 96)

1 2 3 4 510−5

10−4

10−3

10−2

10−1

1

SNR (dB)

BL

ER

N = 144N = 96N = 64N = 48

0 2 4 6 810−5

10−4

10−3

10−2

10−1

1

SNR (dB)

BL

ER

m = 63m = 42m = 28m = 14

(c) BLER with N (m = 42, µ∗ = 0.7) (d) BLER with m (N = 96, µ∗ = 0.7)

Fig. 4: BLER performance of SVC-encoded packet from (17)

where the equality follows from (see Appendix B)

〈φk

‖φk ‖2,φl〉 =

‖h‖2 for k = l‖h‖2µkl for k , l

.

(21)

Let zp = <(

φTp

‖φp ‖2v), then

<〈φp

‖φp ‖2, r0〉 = ‖h‖2 + zp . (22)

In a similar way, we have

<〈φi

‖φi ‖2, r0〉 = ‖h‖2µip + zi, (23)

and hence

P(<〈 φp

‖φp ‖2, r0〉

≥ <〈 φi

‖φi ‖2, r0〉

)= P

(‖h‖2 + zp ≥ ‖h‖2µil + zi

)(a)= P

(‖h‖2 + zp >

‖h‖2µip + zi) P

(‖h‖2 + zp > 0

)+P

(−‖h‖2 − zp >

‖h‖2µip + zi) P

(‖h‖2 + zp < 0

)≥ P

(‖h‖2 + zp > ‖h‖2 |µip | + |zi |

)P

(‖h‖2 + zp > 0

)≥ P

(‖h‖2 + zp > µ∗‖h‖2 + |zi |

)P

(‖h‖2 + zp > 0

),

(24)

where (a) follows from

P(|A| ≥ |B|) = P(A> |B |)P(A > 0)+P(−A > |B |)P (A < 0) .(25)

Since zi ∼ N(0,σ2

v

2 ) from Lemma 1, the second term in (24)is lower bounded as

P(‖h‖2 + zp > 0

)= P

(zp > −‖h‖2

)= 1 −Q

(−‖h‖2σv√

2

)≥ 1 − exp

(−‖h‖22σ2v

), (26)

where the last inequality follows from Q(x) ≤ exp(− x2

2

). In

a similar way, the first term in (24) is lower bounded as

P(‖h‖2 + zp > µ∗‖h‖2 + |zi |

)= 1 − P

(|zi | − zp ≥ (1 − µ∗)‖h‖2

)= 1 − P

(zi − zp ≥ (1 − µ∗)‖h‖2

)P (zi > 0)


−P(−zi − zp ≥ (1 − µ∗)‖h‖2

)P (zi < 0)

(a)= 1 − 2P

(zi − zp ≥ (1 − µ∗)‖h‖2

)P (zi > 0)

(b)≥ 1 −Q

(−‖h‖2(1 − µ∗)

σv

)≥ 1 − exp

(−‖h‖22 (1 − µ

∗)2

2σv2

), (27)

where (a) is because −zi ∼ N(0,σ2

v

2 ) and (b) is because zi −zp ∼ N(0, σ2

v ). By plugging (26) and (27) into (24), we have

P(<〈 φp

‖φp ‖2, r0〉

≥ <〈 φi

‖φi ‖2, r0〉

)≥

(1 − exp

(−‖h‖22 (1 − µ

∗)2

2σv2

)) (1 − exp

(−‖h‖22σ2v

))≥ 1 − exp

(−‖h‖22 (1 − µ

∗)2

2σv2

)− exp

(−‖h‖22σ2v

).

(28)

Note that P(S1 |h) in (19) is the success probability in thefirst iteration for a given channel realization h. In order to ob-tain the unconditional probability, we need to take expectationwith respect to the channel h. That is,

P(S1) =

∫P(S1 |h) fh(x)dx = Eh

[P(S1 |h)

]. (29)

Thus,

P(S1)

= Eh

N∏

i=1,i,pP

(<〈 φp

‖φp ‖2, r0〉

≥ <〈 φi

‖φi ‖2, r0〉

) | h=

N∏i=1,i,p

Eh

[P

(<〈 φp

‖φp ‖2, r0〉

≥ <〈 φi

‖φi ‖2, r0〉

) | h]≥

N∏i=1,i,p

Eh

[1 − exp

(−‖h‖22 (1 − µ

∗)2

2σv2

)− exp

(−‖h‖22σ2v

)| h

]=

N∏i=1,i,p

(1−Eh

[exp

(−‖h‖22 (1 − µ

∗)2

2σv2

)|h]−Eh

[exp

(−‖h‖22σ2v

)|h]).

(30)

Since ‖h‖22 follows Chi-squared distribution with 2m DoF (seeLemma 2), we have

Eh

[exp

(−‖h‖22σv

2

)| h

]=

∫ ∞

0exp

(−

xσv

2

)xm−1 exp (−x)(m − 1)!

dx,

=1(

1σ2

v+ 1

)m , (31)

where the equality follows from∫ ∞0 xn exp (−ax)dx = n!

an+1

for n = 0, 1, 2, ..., a > 0.

In a similar way, we have

Eh

[exp

(−‖h‖22 (1 − µ

∗)2

2σv2

)| h

]=

(1 +(1 − µ∗)2

σ2v

)−m. (32)

Finally, by plugging (31) and (32) into (30), we obtain thelower bound of P(S1) as

P(S1) = Eh

N∏

i=1,i,pP

(<〈 φp

‖φp ‖2, r0〉

≥ <〈 φi

‖φi ‖2, r0〉

) | h=

N∏i=1,i,p

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m)≥

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m)N−1

.(33)

We now move to the success probability for the seconditeration when the first iteration is successful.

Lemma 4. The probability that the support element is chosenat the second iteration under the condition that the firstiteration is successful satisfies

P(S2 |S1

)≥

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m)N−2

.(34)

Proof: When the first iteration is successful, the residualr1 can be expressed as

r1 = r0 − ΦΩ1ss1

(a)= r0 − φpsp= φqsq + v, (35)

where (a) is because the transmit symbols are known inadvance (s1 = sp). After taking similar steps to Lemma 3,one can show that P

(S2 |S1) satisfies (we skip the detailed

steps for brevity)

P(S2 |S1

)= P

(=〈 φq

‖φq ‖2, r1〉

≥ maxi

=〈 φi

‖φi ‖2, r1〉

)=

N∏i=1,i,p,q

P(=〈 φq

‖φq ‖2, r1〉

≥ =〈 φi

‖φi ‖2, r1〉

)≥

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m)N−2

.(36)

It is worth mentioning that the lower bounds of P(S1) and

P(S2 |S1) have the same form except for the exponent. We

are now ready to prove the main theorem.Proof of Theorem 1: By combining Lemma 3 and 4, we

can obtain the lower bound of the success probability Psucc

as

Psucc = P(S2 |S1

)P

(S1

)= P

(=〈 φq

‖φq ‖2, r1〉

≥ maxi

=〈 φi

‖φi ‖2, r1〉

)P

(<〈 φp

‖φp ‖2, r0〉

≥ maxi

<〈 φi

‖φi ‖2, r0〉

)=

N∏i=1,i,p,q

P(=〈 φq

‖φq ‖2, r1〉

≥ =〈 φi

‖φi ‖2, r1〉

)


N∏i=1,i,p

P(<〈 φp

‖φp ‖2, r0〉

≥ <〈 φi

‖φi ‖2, r0〉

)≥

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m) (N−2)+(N−1)

≥

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m)2N

,(37)

which completes the proof.Finally, we present the decoding success probability bound

for general sparsity K .

Theorem 2. The probability that the SVC-encoded packet canbe successfully decoded for a given K satisfies

Psucc &

(1 −

(1 +(1 − µ∗)2

σ2v

)−m)KN

.(38)

Proof: The success probability Psucc is expressed as

Psucc = P(S1,S2, · · · ,SK

)= P

(SK |SK−1, · · · ,S1

)· · · P

(S2 |S1

)P

(S1

)≥

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m)(N−K)+· · ·+(N−1)≥

(1 −

(1 +(1 − µ∗)2

σ2v

)−m−

(1 +

1σ2v

)−m)KN

.(39)

Since the proof is similar to the proof of Theorem 1, we skipthe detailed steps.

If m 1, we approximately have

Psucc &

(1 −

(1 +(1 − µ∗)2

σ2v

)−m)KN

.(40)

It is clear from (40) that the decoding success probabilitydecreases when the information vector is less sparse (i.e., Kis large), which matches with our expectation.

V. IMPLEMENTATION ISSUES

In this section, we discuss the implementation issues in-cluding codebook design, high-order modulation, diversitytransmission, pilot-less transmission, and threshold selectionto prevent the false alarm event.

A. Codebook Design

From our analysis in the previous section, we clearlysee that a codebook with small correlation is important toimprove the decoding success probability. As mentioned, asm increases, the correlation between two randomly generatedcodewords decreases, and thus we can basically use any kindof randomly generated sequence. For example, if we usethe Bernoulli random matrix, then the maximum correlationsatisfies µ∗ ≤

√4m−1 ln N

δ with probability exceeding 1 − δ2

[24].Instead of relying on the random sequence, we can alterna-

tively use the deterministic sequences. Well-known determin-istic sequences include chirps, BCH, DFT, and second-order

Reed-Muller (SORM) sequences [25]. For example, SORMis a sequence designed to generate low correlation sequences.SORM of length 2m is defined as

φP,b(a) =(−1)w(b)√

2pi(2b+Pa)T a, (41)

where P is a d × d binary symmetric matrix, a =

[a0 a1 · · · ad−1]T and b = [b0 b1 · · · bd−1]

T are binaryvectors in Zd2 , and w(b) is the weight (number of ones) of b.The corresponding SORM matrix can be expressed as

Φrm =[UP1 UP2 · · · UP2d(d−1)/2

], (42)

where UP j is the 2d × 2d orthogonal matrix whose columnsare the SORM sequences. The maximum correlation ν∗ of theSORM sequence is

ν∗ =

1√2l, l = rank(Pi − Pj)

1√m, l = d

.

(43)

For example, if m = 64 and l = d, then ν∗ = 0.125. The benefitof using SORM sequence is that the correlation betweenany two codewords is a constant and thus the performancevariation can be minimized.

B. High-order Modulation

Since the ensuring reliability is the top priority in URLLC,QPSK modulation would be the popular option in practice.In order to use the QPSK modulation in SVC, we set oneof the nonzero entries in s to 1 and the other to j. Forexample, if the nonzero positions are 5 and 7, then we sets = [0 0 0 0 1 0 j 0 0 0]T and thus the transmit vector x canbe expressed as

x = 1c5 + jc7. (44)

From (44), we can easily see that elements of the trans-mit vector x are mapped to the QPSK symbol (i.e., xi ∈1+ j, 1- j, -1+ j, -1- j). It is worth mentioning that one ad-ditional bit can be encoded by differentiating two possiblechoices (i.e., [1, j] and [ j, 1]). However, this choice willincrease the computational overhead of the decoding algorithmand also degrade the performance little bit. When the highersparsity is used, this mapping can be readily extended to thehigh order modulation (e.g., K = 4 for 16-QAM and K = 6for 64-QAM). Specifically, if K = 4, we map the element in xto the 16-QAM symbol by setting two nonzero entries to 1, 2and the remaining nonzero entries to j, 2 j. In a similar way, ifK = 6, then we can transmit 64-QAM symbols by setting threeof the nonzero entries to 1, 2, 3 and the remaining ones to j,2 j, and 3 j. The normalization factor (α in (3)) corresponding

M-QAM is α =√

2(M−1)3 .

C. Diversity Transmission

One can easily integrate the diversity scheme to SVC tofurther improve the reliability. The first option is to use the fre-quency diversity in which the SVC-encoded packet is repeatedL times in L distinct frequency bands. The benefit of thefrequency diversity is that the diversity gain can be achieved


without increasing the transmission latency. Specifically, byapplying the maximal-ratio combining at the receiver for thesame symbol of the repeated packets, effective SNR can beincreased and thus the BLER performance can be improved[26]. For example, when the SVC-encoded packet is repeatedfor L = 8 times, due to the power gain of the combinedsymbol, the required SNR to achieve the desired URLLCperformance (e.g., 10-5 BLER) can be reduced from 3 dBto 3 − 10 log10(L) = -6dB in AWGN environments. On top ofthe frequency diversity, other diversity schemes such as time,antenna, and space diversity can also be easily incorporated.

D. SVC without Pilot

When the channel is a constant or channel variation is verysmall (i.e., h ≈ const.), which is true for mobile devices understatic or slowly moving environments, decoding of the SVCpacket can be performed without pilot transmission, resultingin a substantial reduction of the resources, transmission power,receiver processing time, and also implementation cost. In fact,since the packet length is smaller than the channel coherencetime, this assumption holds true in many realistic scenarios.Pilot-less transmission is done by slightly modifying thesystem model such that the system matrix equals the codebookC and the sparse vector contains the channel component(s′ = hs). That is,

y = HCs + v= Cs

′

+ v

=

c1 . . . cN

hs1...

hsN

+v1...vm

. (45)

Recalling that the goal of the SVC decoding is to find out thenonzero positions of s′ vector, we can perform the decodingwithout the channel knowledge. When the channel variationis flat in the frequency axis, tall packet structure (stretched infrequency axis) is preferred. Whereas, if the channel variationis very small in time-domain, horizontal packet (stretched intime axis) would be a good choice.

E. Threshold to Prevent False Alarm Event

To distinguish the false alarm event from the normal de-coding process, we examine the probability that the residualafter the sparse recovery algorithm is not pure noise. In fact,if the SVC decoding is finished successfully, the residualcontains the noise contribution only (rK = v) so that theresidual power ‖rK ‖22 can be readily modeled as a Chi-squaredrandom variable with 2m degree of freedom. Naturally, onecan reject this hypothesis if the residual power is too largeand lies outside of the pre-defined confidence interval. Inother words, if ‖rK ‖22 > F−1

‖v‖22(1 − Pth) where Pth is the

pre-defined probability threshold (e.g., Pth = 0.01) and F−1‖v‖22

is the inverse cumulative distribution function of Chi-squaredrandom variable, then we declare the hypothesis is not true(i.e., decoding is not successful) and discard the decodedpacket. To evaluate the effectiveness of this thresholding

-2 0 2 4 6 8 10 1210−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Prob

.of

fals

eal

arm

With residual thresholdWith 16-bit CRC

Fig. 5: Decoding failure as a function of SNR (Pth = 10−5).

approach, we simulate the probability of false alarm as afunction of SNR for the conventional 16-bit CRC and theproposed residual-based thresholding. As is clear from Fig.5, the residual-based thresholding performs similarly to theCRC-based error checking.

VI. SIMULATIONS AND DISCUSSIONS

A. Simulation Setup

In this section, we examine the performance of the proposedSVC technique. Our simulation setup is based on the downlinkOFDM system in the 3GPP LTE-Advanced Rel.13 [11]. Asa channel model, we use AWGN and realistic ITU channelmodels including extended typical urban (ETU) and extendedpedestrian-A (EPA) channel model [11]. For comparison, wealso investigate the performance of the conventional PDCCHof LTE-Advanced system, polar code-based PDCCH of 5Gsystems [27], and AWGN lower bound. We test the transmis-sion of b bit information which consists of information bitbi and CRC bit bc . In the conventional PDCCH method, theconvolution code with rate 1

3 with the 16-bit CRC is employed.Since the block size of the polar code is not flexible, we setthe rate 1

4 to test similar conditions (b = 24 and m = 32).In the proposed SVC algorithm, we set the random binaryspreading codebook with N = 96 and K = 2. To ensure thefair comparison, we use the same number of resources (m = 42with L = 8 repetitions) in the control packet transmission. Asa performance measure, we use BLER of the codeblocks.

B. Simulation Results

In Fig. 6(a), we investigate the BLER performance ofthe proposed SVC method and competing schemes underAWGN channel condition. We observe that the proposed SVCtechnique outperforms the conventional PDCCH and polarcode-based scheme, achieving more than 4 dB gain over theconventional PDCCH and about 1.1 dB gain over the polarcode-based scheme at 10-5 BLER point. Even in realisticscenarios such as EPA and EVA channels in LTE-Advanced,


-12 -10 -8 -6 -4 -2 010−5

10−4

10−3

10−2

10−1

1

SNR (dB)

BL

ER

Lower boundSVCPDCCHPolar code

-5 0 5 10 15 20 25 30 3510−5

10−4

10−3

10−2

10−1

1

SNR (dB)

BL

ER

PDCCH (ETU channel)SVC (ETU channel)PDCCH (EPA channel)SVC (EPA channel)

(a) (b)

Fig. 6: BLER performance as a function of SNR (bi = 12, bc = 16, m = 42, L = 8, and N = 96) for (a) AWGN channel and(b) ETU and EPA channel.

-12 -8 -4 0 4 810−5

10−4

10−3

10−2

10−1

1

SNR (dB)

BL

ER

PDCCH (bi = 12) SVC (bi = 12)PDCCH (bi = 24) SVC (bi = 24)PDCCH (bi = 48) SVC (bi = 48)PDCCH (bi = 96) SVC (bi = 96)

Fig. 7: BLER performance for various size of control infor-mation (L = 8).

we observe that the performance gain of the proposed SVCscheme over competing schemes is maintained (see Fig. 6(b)).

In Fig. 7, we evaluate the BLER performance of PDCCHand SVC as a function of SNR for various information bitsize (bi = 12, 24, 48, and 96). These results demonstrate thatthe proposed SVC technique can deliver more information bitsthan the conventional PDCCH can support. For example, SVCcan deliver twice more information than PDCCH in the lowSNR region (for example, bi = 12 of PDCCH and bi = 24of SVC in Fig. 7). To further investigate this, we plot theminimum SNR to achieve the target BLER as a function ofthe information bit size in Fig. 8. For example, to achieve 10-5

BLER with b = 10, it requires -2.9 dB for PDCCH while -6.2dB SNR for SVC, resulting in 3.3 dB gain in performance. Itis worth mentioning that the coding gain of the conventionalPDCCH improves with the codeblock size so that the gap

10 20 40 60 80 100-7

-5

-3

-1

1

3

5

b (bit)

SNR

req

@10−

5B

LE

R

PDCCH (b = bi + bc = 12 + 16)SVC (b = bi = 12)

Fig. 8: Minimum required SNR for achieving 10-5 BLER (m =42, L = 8, and N = 96)

between the SVC and PDCCH diminishes gradually as thenumber of information bits increases.

Next, we evaluate the latency performance of the SVCand PDCCH (see Fig. 9). In this experiments, we plot thedistribution of transmission latency to achieve 10-5 BLERwhen n-repetition scheme is employed. Transmission latency isdefined as the time from the initial transmission to the time thatthe packet is successfully decoded at the mobile terminal.10

These results demonstrate that most of SVC packets satisfythe URLLC requirement (1 ms latency).

Finally, we investigate the performance of SVC in thesmall cell scenarios where the received signal contains aconsiderable amount of interference from adjacent basesta-tions. Note that densely deployed small cell (pico, femto, and

10In our experiments, we ignored the decoding latency.


0.21ms

(n = 1)

0.42ms

(n = 2)

0.64ms

(n = 3)

0.85ms

(n = 4)

1.07ms

(n = 5)

1.28ms

(n = 6)

0

0.2

0.4

0.6

0.8

1

Transmission latency (ms)

Prob

abili

tySVC

PDCCH

Fig. 9: Probability of transmission latency for achieving 10-5

BLER (bi = 12, m = 42, L = 8, N = 96, and SNR=−12 dB).

-12 -9 -6 -3 0 310−5

10−4

10−3

10−2

10−1

1

SINR (dB)

BL

ER

PDCCH without interf.PDCCH with interf.SVC without interf.SVC with interf.

Fig. 10: BLER performance as a function of SINR (bi = 12,m = 42, L = 8, N = 96, and interference power is half of thesignal power).

micro) environments will play a key role to enhance the cellthroughput in 5G and how to manage the interference is thekey to the success of small cell networks. In our simulations,we set the power level of interference to half of the desired cellsignal. Since the SVC transmission is based on the multi-codespreading and also the effective transmit power per symbolis large (see Section III.A), SVC can effectively manage theinterference. Whereas, since the conventional PDCCH hasno such interference protection mechanism, error correctioncapability of PDCCH is degraded significantly and thus thePDCCH performs very poor as shown in Fig. 10.

VII. CONCLUSION

In this paper, we have proposed the short packet transmis-sion strategy for URLLC. The key idea behind the proposedSVC technique is to transform an information vector intothe sparse vector in the transmitter and to exploit the sparserecovery algorithm in the receiver. Metaphorically, SVC canbe thought as a marking dots to the empty table. As long asthe number of dots is small enough and the measurementscontain enough information to figure out the marked cellpositions, accurate decoding of SVC packet can be guaranteed.We demonstrated from the numerical evaluations that theproposed SVC scheme is very effective in URLLC scenarios.In this paper, we restricted our attention to the URLLCscenario but we believe that there are many other applicationssuch as mMTC that the SVC technique can be applied to.Also, there are many interesting extensions and variationsworth investigating, such as the information embedding innonzero positions, channel aware sparse vector coding, andcombination of SVC and error correction codes.

APPENDIX APROOF OF LEMMA 1

Let ui =ai‖ai ‖2 , then it is clear that ui is a random vector

with zero mean and unit variance. Also, let X =aTi a j

‖ai ‖2 , thenX = uT

i aj . One can easily show that X conditioned on anyrealization of ui = u is a standard Gaussian. This is becauseE[X |ui = u] = E[uTaj] = uT E[aj] = 0 and Var(X |ui = u) =E[uTajaTj u] = uTu = 1. Further,

fX (x) =∫u

fX |ui(x |u) fui (u)du

=1√

2πexp

(−

x2

2

) ∫u

fui (u)du

=1√

2πexp

(−

x2

2

), (A.1)

which is the desired result.

APPENDIX BDERIVATION OF (21)

Noting that φi = [h11c1i h22c2i · · · hmmcmi]T and µi j =

φTi φ j

‖h‖22, we have

〈φi

‖φi ‖2,φj〉 =

φTi φj

‖φi ‖2. (B.1)

Since ‖φi ‖2 =√| h11c1i |2 + · · · + | hmmcmi |

2 = ‖h‖2, we have

〈φi

‖φi ‖2,φj〉 = ‖h‖2

φTi φj

‖h‖22= ‖h‖2µi j . (B.2)

In particular, i = j, µi j = 1 and thus

〈φi

‖φi ‖2,φi〉 = ‖h‖2. (B.3)


From (B.2) and (B.3), we have 〈φi

‖φi ‖2,φj〉 =

‖h‖2 for i = j‖h‖2µi j for i , j .

ACKNOWLEDGMENT

The authors would like to thank Prof. Daeyoung Park andMr. Luong Nguyen for their help in the performance analysisin Section IV.

REFERENCES

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Hyoungju Ji (M’16) is currently working towardthe Ph.D. degree at the School of Electrical andComputer Engineering, Seoul National University,Seoul, Korea. He joined Samsung Electronics in2007, and has been involved in 3GPP RAN1 LTE,LTE-Adv., LTE-Adv. Pro, and 5G NR technologydevelopments and standardizations. His current in-terests include multi-antenna techniques, FD-MIMO,massive connectivity, machine type communications,and IoT communications.

Sunho Park (M’16) received the B.S., M.S., andPh.D. degrees from the school of Information andCommunication, Korea University, Seoul, in 2008,2010, and 2015, respectively. From 2015 to 2017, hewas with Institute of New Media and Communica-tions, Seoul National University, Seoul, South Korea,as a Research Assistant Professor. He is currentlya senior engineer with Samsung Electronics. Hisresearch interests include wireless communicationsand signal processing.


Byonghyo Shim (S’95-M’97-SM’09) received theB.S. and M.S. degrees in control and instrumentationengineering from Seoul National University, Korea,in 1995 and 1997, respectively, and the M.S. degreein mathematics and the Ph.D. degree in electricaland computer engineering from the University ofIllinois at UrbanaâASChampaign, USA, in 2004 and2005, respectively. From 1997 and 2000, he was withthe Department of Electronics Engineering, KoreanAir Force Academy, as an Officer (First Lieutenant)and an Academic Full-time Instructor. From 2005

to 2007, he was with Qualcomm Inc., San Diego, CA, USA, as a StaffEngineer. From 2007 to 2014, he was with the School of Information andCommunication, Korea University, Seoul, as an Associate Professor. Since2014, he has been with the Seoul National University, where he is currently aProfessor with the Department of Electrical and Computer Engineering. Hisresearch interests include wireless communications, statistical signal process-ing, compressed sensing, and machine learning. Dr. Shim is an elected memberof the Signal Processing for Communications and Networking TechnicalCommittee of the IEEE Signal Processing Society. He was a recipient ofthe M. E. Van Valkenburg Research Award from the ECE Department of theUniversity of Illinois in 2005, the Hadong Young Engineer Award from IEIEin 2010, and the Irwin Jacobs Award from Qualcomm and KICS in 2016.He has served as an Associate Editor of the IEEE TRANSACTIONS ONSIGNAL PROCESSING, the IEEE TRANSACTIONS ON COMMUNICA-TIONS, the IEEE WIRELESS COMMUNICATIONS LETTERS, and Journalof Communications and Networks, and a Guest Editor of the IEEE JOURNALON SELECTED AREAS IN COMMUNICATIONS.

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